Conventional differential assemblies utilize four interengaged bevel gears that rotate about two orthogonal axes. In contrast, the differential assembly which is associated with this invention is of the type that does not utilize bevel gears, and is generally of the design shown in U.S. Pat. No. 2,859,641, issued Nov. 11, 1958 in the name of Gleasman. This patent is incorporated herein by reference to the extent necessary to provide specific details of the structure of the differential assembly.
This type of differential includes a gear housing, a pair of drive axles received in bores formed in the sides of the housing, and a differential gear arrangement mounted centrally in a main body portion of the housing for driving the axles. The rotatable gear housing body portion includes a flange formed at one end for receiving a ring gear or other means for providing power input to the differential from the drive shaft of the vehicle in a conventional manner. The gear housing typically is provided with a removable cap at its other end.
The gear arrangement includes a pair of helical worm or side gears, coupled to each axle end as drive gears, together with so called balancing or transfer gears associated with each of the side gears and in mesh with each other for transferring and dividing torque between the axle ends. The transfer gears are mounted in pairs within slots, or windows, formed in the main body portion of the housing, and each transfer gear of a pair rotates on an axis of rotation that is substantially parallel to a tangent of the envelope of an associated axle drive gear.
The transfer gears are in reality combination gears, i.e., the middle portion of each gear constitutes a worm wheel portion while the outer ends of the gear are formed with integral spur gear portions. The arrangement is such that, for any given pair of combination gears, the worm wheel portion of a first combination gear meshes with one side gear while the worm wheel portion of a second combination gear meshes with the other side gear, and the spur gear portions of the respective combination gears mesh with each other.
In one example of this type of differential assembly, a set of three combination gears are arranged substantially in a first single plane at approximately 120.degree. intervals about the periphery of each side gear, each of the three combination gears being paired with a combination gear of a second set of three combination gears similarly arranged with respect to the second side gear in a second single plane parallel to the first plane.
The present invention relates to differential assemblies of the type described above, and particularly those in which the side gears have helix angles inclined in the same direction with respect to their axes of rotation. In such assemblies, when power is applied to the differential housing, both side gears are thrust in the same direction along their aligned axes toward one end of the differential gear housing. In this regard, it is to be noted that the side gear helix angles are generally selected so that when power is applied to the differential gear housing to effect forward movement of a vehicle, both side gears are thrust toward the flange end of the housing. This arrangement, however, produces different bias ratios for different directions of relative drive axle rotation. This invention seeks to reduce these differences without significantly affecting overall, or average, bias ratio for the differential assembly.
Prior to explaining the improved aspects of the differential assembly in accordance with this invention, a brief discussion of frictional resistance, bias ratio, and bias ratio imbalance will prove helpful to an understanding of the invention. For ease of discussion, the side gear closest to the flange end of the differential gear housing will be referred to as a "bottom" side gear; the side gear closest to the cap end of the housing will be referred to as a "top" side gear; and the flange end of the differential gear housing is assumed to be on the left side of the housing as viewed from the rear of the vehicle. Moreover, with respect to examples discussed herein, forward motion of the vehicle is assumed.
Bias ratio is a measure of torque distribution between drive axles which can be maintained by relatively rotating drive axles, and is expressed as a quotient of the amount of torque in the drive axle having the most torque divided by the amount of torque in the other drive axle. Bias ratio is produced by frictional resistance in a differential which restricts the transmission of torque between drive axles. Generally, frictional resistance causes a percentage reduction in the amount of torque that is transferable between drive axles. The magnitude of this reduction is proportional to the frictional resistance.
Torque transfer through the differential occurs from the drive axle having the larger amount of torque to the drive axle having the smaller amount of torque. Accordingly, the drive axle which includes the larger amount of torque may also be considered as the "input" axle and the drive axle which includes the smaller amount of torque may be considered as the "output" axle. In this context, the terms input and output refer only to transfers of torque between drive axles and are not related to torque transfers between the differential housing and the drive axles collectively.
It is known, for example, that the drive axle connected to the inside wheel in a turn exerts a greater resistance to rotation of the differential housing than the drive axle connected to the faster rotating outside wheel. Accordingly, in opposite directions of turns, one drive axle is loaded more than the other. That is, in one direction of relative drive axle rotation (e.g., a right turn) the drive axle coupled to the top side gear is loaded more than the drive axle coupled to the bottom side gear and, in the opposite direction of relative drive axle rotation (e.g., left turn), the drive axle coupled to the bottom side gear is loaded more than the drive axle coupled to the top side gear. This differential drive axle loading in opposite directions of relative drive axle rotation may result in different bias ratios being associated with such different directions of drive axle rotation. This condition is termed bias ratio imbalance.
Since the problem is one in which bias ratio is decidedly higher in one direction of relative drive axle rotation than the other, it may be understood that frictional resistance in the differential is greater in the direction of relative drive axle rotation associated with the higher bias. However, the same frictional surfaces are known to be in contact in both directions of relative rotation. Frictional forces generated by frictional surfaces are determined by the coefficients of friction of the respective contacting surfaces and the normal forces applied to the surfaces. Since the frictional properties of the contacting surfaces (i.e., coefficients of friction) do not change between opposite directions of drive axle rotation, it may be further understood that differences in frictional resistance between the opposite directions of drive axle rotation are associated with changes in the normal forces applied to the frictional surfaces.
In other words, in the high bias direction of relative drive axle rotation, normal forces applied to the frictional surfaces in the differential are generally higher than in the low bias direction of relative drive axle rotation. Forces are applied to the frictional surfaces of the differential largely because of reaction forces generated at the mounting surfaces of the gearing in the differential. While there are frictional forces generated at each gear mesh, it is understood that the problem of bias ratio imbalance originates at particular mounting surfaces. It is well known, for example, that in order for gears to transmit power, the gears must be supported with all of their reactions contained. Mounting surfaces within the differential have frictional properties, and the normal forces which are applied against the mounting surfaces are, in fact, reaction forces which are required to contain the differential gearing in its operative position. From the above, it follows that reaction forces within the differential required to contain the gearing are larger in the high bias direction of relative drive axle rotation than in the low bias direction.
Typically, the largest reaction forces generated within the differential are side gear thrust forces. It is known that such thrust forces may be calculated at the point of mesh on the side gears. A first component (tangential) of the force applied at the point of mesh either contributes to the rotation of the side gear or conveys rotation of the side gear to an enmeshed gear member, and a second component (axial) of the applied force thrusts the side gear in a direction along its axis. The ratio of the axial force to the tangential force is equal to the tangent function of the side gear helix angle as given by the equation below: EQU W.sub.x /W.sub.t =tan (psi) (1)
where "W.sub.x " is the axial force, "W.sub.t " is the tangential force and "psi" is the helix angle. Thus, the axial thrust of the side gears which must be constrained by mounting surfaces within the differential may be understood to be a product of the tangential driving force of each side gear multiplied by the tangent function of the respective side gear helix angles.
As may be expected, the reaction surfaces which constrain the axial thrust of the side gears are located opposite the end faces of the side gears. The interface between the bottom side gear and housing is used to constrain the axial movement of both the top side gear and bottom side gear. The interface between side gears is used to constrain axial movement of the top side gear.
The effect of the frictional forces generated at the side gear interfaces is to either (a) decrease the tangential driving load conveyed by a side gear or (b) increase the tangential driving load required to cause its rotation. Keeping in mind that the tangential driving loads associated with the two side gears are related through the side gear helix angle to the frictional resistance to relative drive axle rotation within the differential, it will be understood that the tangential driving loads associated with one direction of relative drive axle rotation are larger than the tangential driving loads associated with the opposite direction of relative drive axle rotation.
The problem may be simplified or reduced to its essential components by considering the differential to be frictionless except at the two side gear interfaces. In accordance with conventional practices, the two side gears may be considered to be equal in diameter and include equal helix angles. Under these conditions, the tangential driving loads of each side gear are equal in magnitude. Thus, it remains to be shown only that in one direction of relative drive axle rotation, the tangential driving loads are greater than in the opposite direction of relative drive axle rotation.
As previously explained, in one direction of relative drive axle rotation, the drive axle coupled to the top side gear is considered as "input" to the differential, i.e., the drive axle connected to the top side gear has the larger amount of torque. In this case, the tangential driving load acting at the mesh of the top side gear is reduced with respect to the load applied to the "input" axle by the amount of resistance to top side gear rotation developed at the interface between side gears.
This may be readily shown by way of an equation by taking advantage of a number of mathematical expedients. First, forces at the mesh of the respective side gears are considered to act at a unit distance from the side gear axis of rotation. This enables the interchange of units of force and torque. Second, the side gear helix angles may be considered equal to forty-five degrees. Since the tangent of forty-five degrees is equal to unity, tangential and axial forces acting at point of mesh of the side gears are equal. All frictional forces at the end faces of the side gears are also assumed to be acting at a unit radius as well. Accordingly, the tangential component of the top (and bottom) side gear mesh is equal to: EQU W.sub.t =A.sub.i -(W.sub.t *u.sub.2) (2)
where A.sub.i is the input load or torque on the drive axle associated with the top side gear and u.sub.2 is the coefficient of friction at the interface between the top side gear and the bottom side gear.
Since frictional forces have been discounted elsewhere in the differential, it is also now possible to express the tangential driving load of the bottom side gear in terms of its relationship with the output axle. The portion of the tangential load of the bottom side gear mesh which is received by the output axle is reduced by friction generated at both the interface between side gears as well as friction generated at the interface between the bottom side gear and the housing. This relationship may be expressed as follows: EQU W.sub.t =A.sub.o +2(W.sub.t *u.sub.1)+(W.sub.t *u.sub.2) (3)
where A.sub.o is the output load or torque on the drive axle associated with the bottom side gear and u.sub.1 is the coefficient of friction at the interface between the bottom side gear and the differential housing. It is now possible to algebraically transform equations (2) and (3) as equalities of each axle torque. EQU A.sub.i =W.sub.t (1+u.sub.2) (4) EQU A.sub.o =W.sub.t (1-2u.sub.1 -u.sub.2) (5)
Thus, the bias ratio when the drive axle associated with the top side gear is considered as "input" may be expressed as follows: ##EQU1##
In the opposite direction of differential rotation, the drive axle associated with the bottom side gear is considered input to the differential. The tangential driving load received by the bottom side gear is reduced by frictional forces acting to restrict bottom side gear rotation at both the interface between the bottom side gear and housing and the interface between side gears. This relationship may be summarized as follows: EQU W.sub.t =A.sub.i -2(W.sub.t *u.sub.1)-(W.sub.t *u.sub.2). (7)
It can now already be seen that the tangential driving load at the side gear mesh is reduced with respect to equation (2) by the term "2(W.sub.t *u.sub.1)." Accordingly, it may be anticipated that frictional forces which resist the transfer of torque between drive axles are larger in the direction of relative drive axle rotation associated with the input axle being coupled to the top side gear.
Continuing the mathematical derivation, the tangential driving load at the side gear mesh may also be expressed in connection with the output axle associated with the top side gear. In this case, the output load or torque is reduced from the tangential driving load at the top side gear mesh by frictional forces generated at the interface between the side gears. This may be mathematically expressed as follows: EQU W.sub.t =A.sub.o +(W.sub.t *u.sub.2). (8)
Algebraic transformation may be used to set equations (7) and (8) equal to respective axle torques as follows: EQU A.sub.i =W.sub.t (1+2u.sub.1 +u.sub.2) (9) EQU A.sub.o =W.sub.t (1-u.sub.2). (10)
Thus, the bias ratio when the drive axle coupled to the bottom side gear is considered as input may be expressed as follows: ##EQU2##
Given that the coefficients of friction do not change between opposite directions of drive axle rotation, it may now be demonstrated by comparison of equations (6) and (11) through the repetition of examples in the ordinary range of coefficient values (e.g., 0.01 to 0.2) that the bias ratio associated with input to the top side gear is larger than the bias ratio associated with input to the bottom side gear. The bias ratios between opposite directions of drive axle rotation tend to become closer in value but lower in magnitude as the coefficients of friction are reduced.
It is already known from U.S. Pat. No. 4,191,071 to reduce the coefficient of friction (u.sub.1) at the interface between the bottom side and housing which has the effect of decreasing the difference between bias ratios associated with opposite directions of differential rotation. However, overall bias ratio is also reduced. This may be undesirable in applications in which higher bias ratios are needed. Further, there are practical limitations relating to cost and the availability of bearings which can sustain anticipated loads which limit the amount the coefficient of friction at this interface can be reduced.
Similar "torque equalizing" thrust bearings are disclosed in U.S. Pat. Nos. 4,491,035 and 2,859,641.